Strongly consistent autoregressive predictors in abstract Banach spaces

This work derives new results on strong consistent estimation and prediction for autoregressive processes of order 1 in a separable Banach space B. The consistency results are obtained for the component-wise estimator of the autocorrelation operator in the norm of the space L(B) of bounded linear op...

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Bibliographic Details
Published in:Journal of multivariate analysis 2019-03, Vol.170, p.186-201
Main Authors: Ruiz-Medina, María D., Álvarez-Liébana, Javier
Format: Article
Language:English
Subjects:
Banach spaces
Continuous embeddings
Functional plug-in predictors
Strongly consistent estimators
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Summary:This work derives new results on strong consistent estimation and prediction for autoregressive processes of order 1 in a separable Banach space B. The consistency results are obtained for the component-wise estimator of the autocorrelation operator in the norm of the space L(B) of bounded linear operators on B. The strong consistency of the associated plug-in predictor then follows in the B-norm. A Gelfand triple is defined through the Hilbert space constructed in Kuelbs’ lemma (Kuelbs, 1970). A Hilbert–Schmidt embedding introduces the Reproducing Kernel Hilbert space (RKHS), generated by the autocovariance operator, into the Hilbert space conforming the Rigged Hilbert space structure. This paper extends the work of Bosq (2000) and Labbas and Mourid (2002).
ISSN:0047-259X
1095-7243
DOI:10.1016/j.jmva.2018.08.001